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Homework 6 Solution

# Homework 6 Solution - Homework 6 Solution P671-Group A-1...

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Homework 6 Solution P671-Group A-1: Part (a) 𝑀?𝑛 : 50000 𝑆 + 100000 𝐹 ? . ? . 5 𝑆 + 17 𝐹 40 20 𝑆 + 7 𝐹 60 𝑆 ≥ 0; 𝐹 ≥ 0 Part (b): No; No Part (c): It accounts for the non-linearity between 𝑆 and 𝐹 . P672-Group A-9: 𝑀?𝑥 : 10 min( 𝑥 1 + 𝑥 2 , 120) 20 𝑥 1 2 40 𝑥 1 3 ? . ? . 0 ≤ 𝑥 1 70 0 ≤ 𝑥 2 70 P679: 1. Convex 2. Neither 3. Convex 4. Concave 5. Concave 6. Neither 7. Convex 8. Concave 9. Concave 10. Convex if 4 ?? ≥ ? 2 ; Neither otherwise. 11. Prove by contradiction. Suppose the local optimal solution is not optimal solution. By the fact that f(x) is convex, it can be shown that the local optimal solution violates the definition of local optimum. 12. Take two points 𝑀 1 and 𝑀 2 from set 𝑆 . For any 0 ≤ 𝛼 ≤ 1 , the convexity of ? ( ) and ? ⋅ imply that ? 𝛼𝑀 1 + 1 − 𝛼 𝑀 2 ≤ 𝛼? 𝑀 1 + 1 − 𝛼 ? ( 𝑀 2 ) and ? 𝛼𝑀 1 + 1 − 𝛼 𝑀 2 ≤ 𝛼? 𝑀 1 + 1 − 𝛼 ? ( 𝑀 2 ) . Summing up these two inequalities yields ? 𝛼𝑀 1 + 1 − 𝛼 𝑀 2 ≤ 𝛼? 𝑀 1 + 1 − 𝛼 ? ( 𝑀 2 )

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Homework 6 Solution - Homework 6 Solution P671-Group A-1...

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